A couple whose infant was killed by a falling tree limb at the Central Park Zoo are battling to keep private a host of medical and social-media records they say are unrelated to the 2010 tragedy.

Their $50 million lawsuit over the death of 6-month-old Gianna Marie Ricciutti against the city, the Wildlife Conservation Society and the Central Park Conservancy has dragged on for more than three years as the defendants seek everything from driving and medical records to credit-card information, eight years worth of e-mails, and full access to “all” of the couple’s social-media accounts, including LinkedIn and even the brain-game site Luminosity, according to court papers.

Karla del Gallo, 39, barely survived when the 18-inch-thick tree branch fell from a 50-year-old honey locust and crashed on her as she cradled Gianna Marie.

The New Jersey couple, who have moved to Maryland, claim the tree was poorly maintained, which led to the horrifying accident.

A judge rejected many of the overreaching requests in a June decision, slamming the demand for access to “all social-media sites” used by del Gallo as “an improper fishing expedition” and noting that the city “offers nothing to support its entitlement to access what is essentially an online brain-game site.”

A lawyer for the Wildlife Conservation Society did not return a message seeking comment.

A spokesman for the city Law Department said the intrusive legal tactics are normal.

“It’s a common practice to request these types of records because they could be relevant to damages issues,” he said.